Is there such a thing as "finite" induction?

Question

I am not sure of the terminology that I am looking for, but I would like to use an inductive proof on the following type of structure. I have something of the form, for every $n \geq 2$ and for any $1 \leq d \leq n-1$, some property $P(n, d)$ is true. I chose to do induction on $d$ since it appears to make the proof simpler (fewer cases in the case-by-case analysis for my problem). So my base-case was $P(n, 1)$, then I showed that $P(n, d-1)$ implies $P(n, d)$ for any $2 \leq d \leq n-1$. Typically, induction has this "infinite domino effect", like in the proof that $\sum_{i=1}^n i = n(n+1)/2$, but in my proof, this is not the case as the "domino effect" stops at $d = n-1$. Is this an okay thing to do? Do I still call this a proof by induction? Or does it have a different name?

Answer 1

If you have $n$ as a variable (not as a fixed constant like $n=30$) then I am quite sure (though I did to try to write down the details) that you can prove the full induction principle (for natural numbers) from the validity of your special induction principle.

From this point of view, I would really call it a proof by induction.

To see that your special induction principle is a special case of the general induction principle (for natural numbers) is simple: Given a natural number $n$, just apply the general induction principle to the property $$``\text{$d\ge n$ or $P(n,d)$``}$$